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Simulating Rayleigh Streaming and Heat Transfer in a Standing-Wave Thermoacoustic Engine via a Thermal Lattice Boltzmann Method

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Abstract

The streaming flow and convective heat transfer in a standing-wave thermoacoustic engine (SWTAE) filled with helium gas were numerically handled. The mathematical model depicting the flow and heat transfer occurring consists of the extended Brinkman–Forchheimer–Darcy equations under Boussinesq approximation and completed by the temperature equation based on local thermal equilibrium assumption. Their numerical resolution was performed using a thermal lattice Boltzmann method (TLBM) with the approximation of Bhatnagar–Gross–Krook (BGK) implemented in an in-house solver. Such an approach is further validated by a previous study available in the literature with good agreement. The phase variation of streaming velocity, the convective heat effect on Rayleigh streaming, and temperature gradient and porosity effects on SWTAE thermal efficiency have been investigated and amply commented on. It turned out that the convection effect on the acoustic velocity is mainly observed on the engine core hot side while it is negligible on the cold side. Likewise, its influence on the Rayleigh streaming is particularly detected at low thermal gradient and that the minimization of the Rayleigh effect improves the thermoacoustic conversion at large thermal gradient. On the other hand, such efficiency is improved with increasing the core porosity. Based on the findings obtained, the TLBM approach adopted seems suitable to predict such flow's behavior. Thereby, the present work opens up a new course to model and characterize the flow and transfer of heat by convection in a SWTAE.

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Abbreviations

c :

Lattices speed (m·s−1)

c s :

Sound speed (m·s−1)

c i :

Lattice velocity in direction i

C p :

Specific heat capacity at constant pressure (J·kg−1·K−1)

Da :

Darcy number (\(= K/(\varepsilon H^{2} )\))

d p :

Mean pore diameter \((m)\)

\(\overline{\overline{d}} ,\;\overline{\overline{D}}\) :

Strain-rate tensors

\({E}_{c}\) :

Eckert number (\(= U_{ref}^{2} /(C_{f} .\Delta T_{ref} )\))

\(f_{i}\), \(g_{i}\) :

Distribution functions in direction \(i\)

\(f_{i}^{eq}\), \(g_{i}^{eq}\) :

Equilibrium distribution functions

\({F}_{\varepsilon }\) :

Forchheimer form coefficient

F :

Total body force (N·kg−1)

\(g\) :

Gravity acceleration (m·s−2)

G :

Buoyancy force

H, L :

Characteristic height and length (m)

K :

Porous medium permeability (m2)

k :

Thermal conductivity (W·m−1·K−1)

P :

Pressure (Pa)

p :

Dimensionless pressure

Pr :

Prandtl number (\(= \mu C_{p,f} /k_{f}\))

r :

Reflection coefficient

Ra :

Rayleigh number (\(= g\beta (T_{H} - T_{C} )H^{3} /(\nu \alpha )\))

Rc, R K :

Ratios of heat capacities and thermal conductivities

Re :

Reynolds number (\({\text{Re}} = U_{ref} H/\nu\))

t, t ̃ :

Time (s) and dimensionless time

T:

Temperature (K)

\(\overrightarrow {U} \left( {U,V} \right)\) :

Velocity (m·s−1)

\(\overrightarrow {u} \left( {u,v} \right)\) :

Dimensionless velocity

\(X,\;Y\) :

Cartesian coordinates (m)

\(x,\;y\) :

Dimensionless coordinates

\(Q,\;W\) :

Heat power (W), acoustic power (W)

\(\alpha\) :

Thermal diffusivity

\(\beta\) :

Thermal expansion coefficient (K−1)

\(\varepsilon\) :

Porosity

\(\eta\) :

Thermal efficiency (%)

\(\mu\) :

Dynamic viscosity (Pa·s)

\(\nu\) :

Kinematic viscosity (\(m^{2} \cdot s^{ - 1}\))

\(\Phi\) :

Viscous dissipation

ϕ :

Dimensionless viscous dissipation

\(\vartheta\) :

Binary number

\(\psi\) :

Acoustic wave's phase

\(\rho \) :

Density (kg·m−3)

\(\theta \) :

Dimensionless temperature

\(g\) :

Gravity acceleration (m·s−2)

\(\nabla (..)\) :

Gradient operator

\(\nabla .(..)\) :

Divergence operator

\(\nabla^{2} (..)\) :

Laplacian operator

\(\Delta (..)\) :

Difference operator

a, C, H :

Ambient, cold, hot

\(e,f, s\) :

Effective, fluid, solid

I :

Initial state

p :

Porous medium

ref :

Reference

BCs:

Boundary conditions

BGK:

Bhatnagar–Gross–Krook

BFD:

Brinkman–Forchheimer–Darcy

CFD:

Computational fluid dynamics

LBE:

Lattice Boltzmann equation

LBM:

Lattice Boltzmann method

LTE:

Local thermal equilibrium

SWTAE:

Standing-wave thermoacoustic engine

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Acknowledgements

The first author would like to thank Dr. Riheb Mabrouk (LESTE/ENIM) for the many fruitful discussions and helpful comments when revising the manuscript. Likewise, the authors would like to thank the anonymous reviewers for their thoughtful comments and worthwhile suggestions that have helped to further improve this paper.

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  1. Laboratoire d’études des Systèmes Thermiques et Énergétiques (LESTE), Ecole National d’Ingénieurs de Monastir (ENIM), Université de Monastir, 5000, Monastir, Tunisia

    Sahar Slimene, Abir Yahya & Hacen Dhahri

  2. Laboratoire de Génie Civil et Géo-Environnement (LGCgE), Univ. Artois, IMT Nord Europe, Junia, Univ. Lille, ULR 4515, Technoparc Futura, 62400, Béthune, France

    Hassane Naji

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Slimene, S., Yahya, A., Dhahri, H. et al. Simulating Rayleigh Streaming and Heat Transfer in a Standing-Wave Thermoacoustic Engine via a Thermal Lattice Boltzmann Method. Int J Thermophys 43, 100 (2022). https://doi.org/10.1007/s10765-022-03016-x

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